Solving initial value problems for stiff or non-stiff systems of first-order ordinary differential equations (ODEs).

The R function `lsoda`

provides an interface to the FORTRAN ODE
solver of the same name, written by Linda R. Petzold and Alan
C. Hindmarsh.

The system of ODE's is written as an R function (which may, of
course, use `.C`

, `.Fortran`

,
`.Call`

, etc., to call foreign code) or be defined in
compiled code that has been dynamically loaded. A vector of
parameters is passed to the ODEs, so the solver may be used as part of
a modeling package for ODEs, or for parameter estimation using any
appropriate modeling tool for non-linear models in R such as
`optim`

, `nls`

, `nlm`

or
`nlme`

`lsoda`

differs from the other integrators (except `lsodar`

)
in that it switches automatically between stiff and nonstiff methods.
This means that the user does not have to determine whether the
problem is stiff or not, and the solver will automatically choose the
appropriate method. It always starts with the nonstiff method.

```
lsoda(y, times, func, parms, rtol = 1e-6, atol = 1e-6,
jacfunc = NULL, jactype = "fullint", rootfunc = NULL,
verbose = FALSE, nroot = 0, tcrit = NULL,
hmin = 0, hmax = NULL, hini = 0, ynames = TRUE,
maxordn = 12, maxords = 5, bandup = NULL, banddown = NULL,
maxsteps = 5000, dllname = NULL, initfunc = dllname,
initpar = parms, rpar = NULL, ipar = NULL, nout = 0,
outnames = NULL, forcings = NULL, initforc = NULL,
fcontrol = NULL, events = NULL, lags = NULL,...)
```

y

the initial (state) values for the ODE system. If `y`

has a name attribute, the names will be used to label the output
matrix.

times

times at which explicit estimates for `y`

are
desired. The first value in `times`

must be the initial time.

func

either an R-function that computes the values of the
derivatives in the ODE system (the *model definition*) at time
t, or a character string giving the name of a compiled function in a
dynamically loaded shared library.

If `func`

is an R-function, it must be defined as:
`func <- function(t, y, parms,...)`

. `t`

is the current
time point in the integration, `y`

is the current estimate of
the variables in the ODE system. If the initial values `y`

has
a `names`

attribute, the names will be available inside `func`

.
`parms`

is a vector or list of parameters; ... (optional) are
any other arguments passed to the function.

The return value of `func`

should be a list, whose first
element is a vector containing the derivatives of `y`

with
respect to `time`

, and whose next elements are global values
that are required at each point in `times`

. The derivatives
must be specified in the **same order** as the state variables `y`

.

If `func`

is a string, then `dllname`

must give the name
of the shared library (without extension) which must be loaded
before `lsoda()`

is called. See package vignette
`"compiledCode"`

for more
details.

parms

vector or list of parameters used in `func`

or
`jacfunc`

.

rtol

relative error tolerance, either a scalar or an array as
long as `y`

. See details.

atol

absolute error tolerance, either a scalar or an array as
long as `y`

. See details.

jacfunc

if not `NULL`

, an R function, that computes the
Jacobian of the system of differential equations
\(\partial\dot{y}_i/\partial y_j\), or
a string giving the name of a function or subroutine in
`dllname`

that computes the Jacobian (see vignette
`"compiledCode"`

for more about this option).

In some circumstances, supplying
`jacfunc`

can speed up the computations, if the system is
stiff. The R calling sequence for `jacfunc`

is identical to
that of `func`

.

If the Jacobian is a full matrix, `jacfunc`

should return a
matrix \(\partial\dot{y}/\partial y\), where the ith row contains the derivative of
\(dy_i/dt\) with respect to \(y_j\), or a vector containing the
matrix elements by columns (the way R and FORTRAN store matrices).

If the Jacobian is banded, `jacfunc`

should return a matrix
containing only the nonzero bands of the Jacobian, rotated
row-wise. See first example of lsode.

jactype

the structure of the Jacobian, one of `"fullint"`

,
`"fullusr"`

, `"bandusr"`

or `"bandint"`

- either
full or banded and estimated internally or by user.

rootfunc

if not `NULL`

, an R function that computes the
function whose root has to be estimated or a string giving the name
of a function or subroutine in `dllname`

that computes the root
function. The R calling sequence for `rootfunc`

is identical
to that of `func`

. `rootfunc`

should return a vector with
the function values whose root is sought. When `rootfunc`

is
provided, then `lsodar`

will be called.

verbose

if `TRUE`

: full output to the screen, e.g. will
print the `diagnostiscs`

of the integration - see details.

nroot

only used if `dllname`

is specified: the number of
constraint functions whose roots are desired during the integration;
if `rootfunc`

is an R-function, the solver estimates the number
of roots.

tcrit

if not `NULL`

, then `lsoda`

cannot integrate
past `tcrit`

. The FORTRAN routine `lsoda`

overshoots its
targets (times points in the vector `times`

), and interpolates
values for the desired time points. If there is a time beyond which
integration should not proceed (perhaps because of a singularity),
that should be provided in `tcrit`

.

hmin

an optional minimum value of the integration stepsize. In
special situations this parameter may speed up computations with the
cost of precision. Don't use `hmin`

if you don't know why!

hmax

an optional maximum value of the integration stepsize. If
not specified, `hmax`

is set to the largest difference in
`times`

, to avoid that the simulation possibly ignores
short-term events. If 0, no maximal size is specified.

hini

initial step size to be attempted; if 0, the initial step size is determined by the solver.

ynames

logical, if `FALSE`

: names of state variables are not
passed to function `func`

; this may speed up the simulation especially
for large models.

maxordn

the maximum order to be allowed in case the method is
non-stiff. Should be <= 12. Reduce `maxord`

to save storage space.

maxords

the maximum order to be allowed in case the method is stiff. Should be <= 5. Reduce maxord to save storage space.

bandup

number of non-zero bands above the diagonal, in case the Jacobian is banded.

banddown

number of non-zero bands below the diagonal, in case the Jacobian is banded.

maxsteps

maximal number of steps per output interval taken by the solver.

dllname

a string giving the name of the shared library
(without extension) that contains all the compiled function or
subroutine definitions refered to in `func`

and
`jacfunc`

. See package vignette `"compiledCode"`

.

initfunc

if not `NULL`

, the name of the initialisation function
(which initialises values of parameters), as provided in
`dllname`

. See package vignette `"compiledCode"`

.

initpar

only when `dllname`

is specified and an
initialisation function `initfunc`

is in the dll: the
parameters passed to the initialiser, to initialise the common
blocks (FORTRAN) or global variables (C, C++).

rpar

only when `dllname`

is specified: a vector with
double precision values passed to the dll-functions whose names are
specified by `func`

and `jacfunc`

.

ipar

only when `dllname`

is specified: a vector with
integer values passed to the dll-functions whose names are specified
by `func`

and `jacfunc`

.

nout

only used if `dllname`

is specified and the model is
defined in compiled code: the number of output variables calculated
in the compiled function `func`

, present in the shared
library. Note: it is not automatically checked whether this is
indeed the number of output variables calculated in the dll - you have
to perform this check in the code. See package vignette
`"compiledCode"`

.

outnames

only used if `dllname`

is specified and
`nout`

> 0: the names of output variables calculated in the
compiled function `func`

, present in the shared library.
These names will be used to label the output matrix.

forcings

only used if `dllname`

is specified: a list with
the forcing function data sets, each present as a two-columned matrix,
with (time,value); interpolation outside the interval
[min(`times`

), max(`times`

)] is done by taking the value at
the closest data extreme.

See forcings or package vignette `"compiledCode"`

.

initforc

if not `NULL`

, the name of the forcing function
initialisation function, as provided in
`dllname`

. It MUST be present if `forcings`

has been given a
value.
See forcings or package vignette `"compiledCode"`

.

fcontrol

A list of control parameters for the forcing functions.
See forcings or vignette `compiledCode`

.

events

A matrix or data frame that specifies events, i.e. when the value of a state variable is suddenly changed. See events for more information.

lags

...

additional arguments passed to `func`

and
`jacfunc`

allowing this to be a generic function.

A matrix of class `deSolve`

with up to as many rows as elements
in `times`

and as many columns as elements in `y`

plus the number of "global"
values returned in the next elements of the return from `func`

,
plus and additional column for the time value. There will be a row
for each element in `times`

unless the FORTRAN routine `lsoda'
returns with an unrecoverable error. If `y`

has a names
attribute, it will be used to label the columns of the output value.

All the hard work is done by the FORTRAN subroutine `lsoda`

,
whose documentation should be consulted for details (it is included as
comments in the source file `src/opkdmain.f`

). The implementation
is based on the 12 November 2003 version of lsoda, from Netlib.

`lsoda`

switches automatically between stiff and nonstiff
methods. This means that the user does not have to determine whether
the problem is stiff or not, and the solver will automatically choose
the appropriate method. It always starts with the nonstiff method.

The form of the **Jacobian** can be specified by `jactype`

which can
take the following values:

- "fullint"
a full Jacobian, calculated internally by lsoda, the default,

- "fullusr"
a full Jacobian, specified by user function

`jacfunc`

,- "bandusr"
a banded Jacobian, specified by user function

`jacfunc`

the size of the bands specified by`bandup`

and`banddown`

,- "bandint"
banded Jacobian, calculated by lsoda; the size of the bands specified by

`bandup`

and`banddown`

.

If `jactype`

= "fullusr" or "bandusr" then the user must supply a
subroutine `jacfunc`

.

The following description of **error control** is adapted from the
documentation of the lsoda source code
(input arguments `rtol`

and `atol`

, above):

The input parameters `rtol`

, and `atol`

determine the error
control performed by the solver. The solver will control the vector
**e** of estimated local errors in **y**, according to an
inequality of the form max-norm of ( **e**/**ewt** ) \(\leq\) 1, where **ewt** is a vector of positive error weights. The
values of `rtol`

and `atol`

should all be non-negative. The
form of **ewt** is:

$$\mathbf{rtol} \times \mathrm{abs}(\mathbf{y}) + \mathbf{atol}$$

where multiplication of two vectors is element-by-element.

If the request for precision exceeds the capabilities of the machine,
the FORTRAN subroutine lsoda will return an error code; under some
circumstances, the R function `lsoda`

will attempt a reasonable
reduction of precision in order to get an answer. It will write a
warning if it does so.

The diagnostics of the integration can be printed to screen
by calling `diagnostics`

. If `verbose`

= `TRUE`

,
the diagnostics will written to the screen at the end of the integration.

See vignette("deSolve") for an explanation of each element in the vectors containing the diagnostic properties and how to directly access them.

**Models** may be defined in compiled C or FORTRAN code, as well as
in an R-function. See package vignette `"compiledCode"`

for details.

More information about models defined in compiled code is in the package vignette ("compiledCode"); information about linking forcing functions to compiled code is in forcings.

Examples in both C and FORTRAN are in the `dynload`

subdirectory
of the `deSolve`

package directory.

Hindmarsh, Alan C. (1983) ODEPACK, A Systematized Collection of ODE
Solvers; in p.55--64 of Stepleman, R.W. et al.[ed.] (1983)
*Scientific Computing*, North-Holland, Amsterdam.

Petzold, Linda R. (1983) Automatic Selection of Methods for Solving
Stiff and Nonstiff Systems of Ordinary Differential Equations.
*Siam J. Sci. Stat. Comput.* **4**, 136--148.

Netlib: http://www.netlib.org

`lsode`

, which can also find a root`lsodes`

,`lsodar`

,`vode`

,`daspk`

for other solvers of the Livermore family,`ode`

for a general interface to most of the ODE solvers,`ode.band`

for solving models with a banded Jacobian,`ode.1D`

for integrating 1-D models,`ode.2D`

for integrating 2-D models,`ode.3D`

for integrating 3-D models,

`diagnostics`

to print diagnostic messages.

# NOT RUN { ## ======================================================================= ## Example 1: ## A simple resource limited Lotka-Volterra-Model ## ## Note: ## 1. parameter and state variable names made ## accessible via "with" function ## 2. function sigimp accessible through lexical scoping ## (see also ode and rk examples) ## ======================================================================= SPCmod <- function(t, x, parms) { with(as.list(c(parms, x)), { import <- sigimp(t) dS <- import - b*S*P + g*C #substrate dP <- c*S*P - d*C*P #producer dC <- e*P*C - f*C #consumer res <- c(dS, dP, dC) list(res) }) } ## Parameters parms <- c(b = 0.0, c = 0.1, d = 0.1, e = 0.1, f = 0.1, g = 0.0) ## vector of timesteps times <- seq(0, 100, length = 101) ## external signal with rectangle impulse signal <- as.data.frame(list(times = times, import = rep(0,length(times)))) signal$import[signal$times >= 10 & signal$times <= 11] <- 0.2 sigimp <- approxfun(signal$times, signal$import, rule = 2) ## Start values for steady state y <- xstart <- c(S = 1, P = 1, C = 1) ## Solving out <- lsoda(xstart, times, SPCmod, parms) ## Plotting mf <- par("mfrow") plot(out, main = c("substrate", "producer", "consumer")) plot(out[,"P"], out[,"C"], type = "l", xlab = "producer", ylab = "consumer") par(mfrow = mf) ## ======================================================================= ## Example 2: ## from lsoda source code ## ======================================================================= ## names makes this easier to read, but may slow down execution. parms <- c(k1 = 0.04, k2 = 1e4, k3 = 3e7) my.atol <- c(1e-6, 1e-10, 1e-6) times <- c(0,4 * 10^(-1:10)) lsexamp <- function(t, y, p) { yd1 <- -p["k1"] * y[1] + p["k2"] * y[2]*y[3] yd3 <- p["k3"] * y[2]^2 list(c(yd1, -yd1-yd3, yd3), c(massbalance = sum(y))) } exampjac <- function(t, y, p) { matrix(c(-p["k1"], p["k1"], 0, p["k2"]*y[3], - p["k2"]*y[3] - 2*p["k3"]*y[2], 2*p["k3"]*y[2], p["k2"]*y[2], -p["k2"]*y[2], 0 ), 3, 3) } ## measure speed (here and below) system.time( out <- lsoda(c(1, 0, 0), times, lsexamp, parms, rtol = 1e-4, atol = my.atol, hmax = Inf) ) out ## This is what the authors of lsoda got for the example: ## the output of this program (on a cdc-7600 in single precision) ## is as follows.. ## ## at t = 4.0000e-01 y = 9.851712e-01 3.386380e-05 1.479493e-02 ## at t = 4.0000e+00 y = 9.055333e-01 2.240655e-05 9.444430e-02 ## at t = 4.0000e+01 y = 7.158403e-01 9.186334e-06 2.841505e-01 ## at t = 4.0000e+02 y = 4.505250e-01 3.222964e-06 5.494717e-01 ## at t = 4.0000e+03 y = 1.831975e-01 8.941774e-07 8.168016e-01 ## at t = 4.0000e+04 y = 3.898730e-02 1.621940e-07 9.610125e-01 ## at t = 4.0000e+05 y = 4.936363e-03 1.984221e-08 9.950636e-01 ## at t = 4.0000e+06 y = 5.161831e-04 2.065786e-09 9.994838e-01 ## at t = 4.0000e+07 y = 5.179817e-05 2.072032e-10 9.999482e-01 ## at t = 4.0000e+08 y = 5.283401e-06 2.113371e-11 9.999947e-01 ## at t = 4.0000e+09 y = 4.659031e-07 1.863613e-12 9.999995e-01 ## at t = 4.0000e+10 y = 1.404280e-08 5.617126e-14 1.000000e+00 ## Using the analytic Jacobian speeds up execution a little : system.time( outJ <- lsoda(c(1, 0, 0), times, lsexamp, parms, rtol = 1e-4, atol = my.atol, jacfunc = exampjac, jactype = "fullusr", hmax = Inf) ) all.equal(as.data.frame(out), as.data.frame(outJ)) # TRUE diagnostics(out) diagnostics(outJ) # shows what lsoda did internally # }

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